How can I sort a DataGridTemplateColumn on a WPF Toolkit DataGrid?

You need to set the SortMemberPath property on Controls:DataGridTemplateColumn to the name of the CLR property you want to sort on.

Added SortMemberPath="Name" to DataGridTemplateColumn. It started sorting.

<data:DataGridTemplateColumn Header="Name" SortMemberPath="Name" CanUserSort="True">

 <DataGridTemplateColumn SortMemberPath="DataDiNascita" Header="Data di nascita" IsReadOnly="True">
                <DataGridTemplateColumn.CellEditingTemplate>
                    <DataTemplate>
                        <DatePicker SelectedDate="{Binding Path=DataDiNascita}"></DatePicker>
                    </DataTemplate>
                </DataGridTemplateColumn.CellEditingTemplate>
                <DataGridTemplateColumn.CellTemplate>
                    <DataTemplate>
                        <TextBlock Text="{Binding Path=DataDiNascita,StringFormat=\{0:dd/MM/yyyy\}}"></TextBlock>
                    </DataTemplate>
                </DataGridTemplateColumn.CellTemplate>
            </DataGridTemplateColumn>